What dialogue? Apparently, if we’re too stupid to recognize what the question was, DoronShadmi is too important to explain it.

The question is very simple:

What is the minimal logical condition that is needed in order to define the Multitude ?

I’m going to take an uneducated stab at answering this question. First, what I believe is the concluding (hopefully not a conclusion looking for evidence to dogmatically) ideal is possibly a mathematical model to which helps to explain biological systems or systems that govern the laws recognized in the universe. Second, I say this because a great deal of “complexity” programs are computational and imply a randomness that is then correlated to biological system that appear to have random, but recognizable pattern structures. So, in answering the question, I will assume one may be implying a mathematical model for examining a rationale for universal consciousness.

Only kidding.. But, I do wonder if this will turn into a conversation about getting “something” from “nothing”.

Consistency is the most important property of an interesting mathematical framework, because in an inconsistent framework we can prove X and its negation. In that case, the whole idea of proof is meaningless. Classical Mathematics’s consistency is based on XOR connective, where concepts like True and False prevent each other (If X is True, then it cannot be also False and vice versa). It enables the existence of an interesting framework, which is sufficient enough to prove or disprove assumptions. In addition there are cases which are undefined or cannot be proved or disproved within Classical Mathematics.

We show that in order to avoid a framework that can prove X and its negation, Classical Mathematics uses an hidden assumption, which is based on the ability to compare between X and its negation in a one framework in order to conclude that we prove, disprove, or cannot prove or disprove some assumption.

We think that the ability to compare between X and its negation must not be undefined logically, or in other words, the logical connective of the concept of Framework must be defined. We generalized it by using the minimal truth table of 2-valued logic, and noticed that the answer to this question is based on a NXOR (Not XOR) connective.
NXOR connective, if used, enables us:

1. To define the concept of Framework (or Context, if you wish) logically (we avoid the hidden assumption).

2. As a result we invent/discover a consistent mathematical universe that exists between X and its negation, where the XOR connective Classical Mathematics’s consistency is the particular case where the middle (the universe between X and its negation) is excluded.

3. From a NXOR connective point of view both NXOR\XOR connectives complement each other and Complementary Logic (where Excluded-middle is a particular case of it) is a generalization of Classical Logic.
4. XOR is not a hidden assumption from a NXOR point of view.
5. NXOR is a hidden assumption from a XOR point of view.

6. NXOR and XOR contradict each other form a XOR point of view.

7. NXOR and XOR complement each other form a NXOR point of view.

Let Z be NXOR\XOR Logic.

The truth table of Z is:
0 0 → T (NXOR)
0 1 → T (XOR)

1 0 → T (XOR)
1 1 → T (NXOR)

By Z we may fulfill Hilbert’s organic paradigm of the mathematical language. Quoting Hilbert’s famous Paris 1900 lecture:“…The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.”

Am I assuming that we are looking to bridge ‘nothing’ with ‘something’?

Doron Shadmi, Moshe Klein

Let a thing be nothing or something.

Let x be a placeholder of a thing.
Definition 2:
x is called local if for any system A, x is in A xor x is out A returns true.
3
The truth table of locality is:
in out
0 0 F
0 1 T (in , out are not the same) = { }_
1 0 T (in , out are not the same) = {_}
1 1 F

Let x be nothing.

Definition 3:
x is called non-local if for any system A, x is in A nor x is out A returns true.
The truth table of non-locality when x is nothing:
in out
0 0 T (in , out are the same) = { }
0 1 F
1 0 F
1 1 F

Let x be something.

Definition 4:
x is called non-local if for any system A, x is in A and x is out A returns true.
The truth table of non-locality when x is something:
in out
0 0 F
0 1 F
1 0 F
1 1 T (in , out are the same) = {_}_

Let system Z be the complementation between non-locality and locality.

The truth table of Z is:
in out
0 0 T (in , out are the same) = { }
0 1 T (in , out are not the same) = { }_
1 0 T (in , out are not the same) = {_}
1 1 T (in , out are the same) = {_}_

Occam, I didn’t see this before… it looks like a more mathematical version of Howard Bloom’s stuff. I think Gödel, Escher, Bach explains in about six places why things like the XOR/NXOR distinctions don’t make sense (essentially, XOR is an operator within propositional logic, while the idea of proving theorems in steps, or that exactly one of a proposition and its negation is true, is external to the system).

The part about entropy is a New Agened version of existing results about partitions, but even it looks at the most trivial and least interesting portion of it.

Occam, I didn’t see this before… it looks like a more mathematical version of Howard Bloom’s stuff. I think Gödel, Escher, Bach explains in about six places why things like the XOR/NXOR distinctions don’t make sense (essentially, XOR is an operator within propositional logic, while the idea of proving theorems in steps, or that exactly one of a proposition and its negation is true, is external to the system).

The part about entropy is a New Agened version of existing results about partitions, but even it looks at the most trivial and least interesting portion of it.

Glad you’re here, Alon, to clarify this stuff. I’m not familiar with Howard Bloom ... perhaps you could expand a bit on what this all means? Or is it just not worth the time and effort?

Occam, I didn’t see this before… it looks like a more mathematical version of Howard Bloom’s stuff. I think Gödel, Escher, Bach explains in about six places why things like the XOR/NXOR distinctions don’t make sense (essentially, XOR is an operator within propositional logic, while the idea of proving theorems in steps, or that exactly one of a proposition and its negation is true, is external to the system).

Hi Alon,

Are you sure that you understand my work?

If you look at it from a XOR-only point of view, then it does not make sence.

It wants to have its cake and eat it too, and so it is attacking the definition of logical consistency instead of learning some mathematics.

It is a fool.

A dialog that I have found in another forum:

[quote author=“Pavel”]
What exactly did you mean by “the two need not be exclusive”. Please be specific with an example of it not needing to be exclusive and then, for contrast, an example of it needing to be exclusive. I think it’ll be become clear to me as soon as you do it.

[quote author=“AKG”]
In the strictly logical sense, “OR” is not exclusive. So, if I ask, what kind of clothing do you want to buy, and you answer, “a shirt OR a pant,” then you would find it acceptable if we bought just a shirt, just a pant, or both. “XOR” means “exclusive-OR”. If you were to say, “a shirt XOR a pant,” then that means you would want just a shirt, just a pant, but not both. I believe, in most situations, we use “or” as logic uses “XOR.” When someone asks you what you want to eat, and you say, “italian or chinese” you mean either italian, or chinese, but not both. “XOR” is like saying EITHER option 1 OR option 2, BUT not both.

When I say that LEM says that they need not be exclusive, then that means that the case could be that P, ~P, or both. Now you would ask, “both P and ~P? That’s weird.” Indeed, it is, and that’s what LNC is for. LEM says P, ~P, or both, and LNC says, no, not both. Putting them together, we get either P or ~P, just one or the other, not both. Now, I don’t think I can give an example where we have P and ~P. LEM allows it, but LNC does not, so to give an example, I would have to deny LNC. I don’t think I could make a rational English sentence using the word “not” if I were to deny LNC. Technically, such an example is possible, but I don’t think you would buy it as a reasonable example because it would sound so unnatural.

LNC of Classical Logic (the Law of Non-Contradiction) is based on XOR connective of a bivalence framework , and the ability to compare between at least two states is based on a hidden assumption, if we ignore NXOR as the logical basis of the concept of Framework (or Context, if you wish).

Classical Logic is the particular case of NXOR\XOR logic where the middle is excluded.

DrMat does not understand NXOR\XOR Logic because he looks at it from a XOR-only point of view and then he concludes that:
[quote author=“DrMatt”]
It wants to have its cake and eat it too

In NXOR\XOR two opposites define their middle domain, and as a result each product of this domain is a consistent “off spring” of both NXOR and XOR connectives, which complement their middle domain (a domin which is equivalent to the “content” of the Empty-Set from a XOR-only point of view).

In a NXOR\XOR logic the concept of Cake exists in any part of it, and as a result the concept of Cake exists as long as there is some part of it.

In other words Alon, please ask some questions before you conclude something about my work.

Alon - 01 September 2007 02:32 AM

The part about entropy is a New Agened version of existing results about partitions, but even it looks at the most trivial and least interesting portion of it.

Howard Bloom’s basically a crank who writes books about the deep connections between everything and anything. I haven’t actually read him; but he bothered a classmate of mine who was doing math with her boyfriend in public, so that classmate explained to me all the bullshit he told them.

And no, the law of non-contradiction isn’t based on XOR thinking, any more than syllogisms are based on IF-THEN clauses. As Hofstadter explains, the notion that “P and P—> Q imply Q” is a rule in the system of formal logic. It’s true that (P ^ P—> Q)—> Q, where—> means “implies” and ^ means “and,” but that’s a statement within the system, rather than a rule. Otherwise, if it were a rule, you’d have infinite regression, with things like ((P ^ P—> Q) ^ ((P ^ P—> Q)—> Q))—> Q. Likewise, you can’t really express the consistency of the system as something like ~(P ^ ~P) where ~ is negation. That’s a tautology; the trick is to show, system-externally, that the system can’t prove propositions P and ~P at the same time.

Occam, I didn’t see this before… it looks like a more mathematical version of Howard Bloom’s stuff. I think Gödel, Escher, Bach explains in about six places why things like the XOR/NXOR distinctions don’t make sense (essentially, XOR is an operator within propositional logic, while the idea of proving theorems in steps, or that exactly one of a proposition and its negation is true, is external to the system).

The part about entropy is a New Agened version of existing results about partitions, but even it looks at the most trivial and least interesting portion of it.

Glad you’re here, Alon, to clarify this stuff. I’m not familiar with Howard Bloom ... perhaps you could expand a bit on what this all means? Or is it just not worth the time and effort?

I decided to run Howard Bloom’s name through Academic Search Premier. I came up with a book review of his book; Global Brain: The Evolution of Mass Mind from the Big Bang to the 21st Century, done by Michael Shermer. I also checked out Bloom’s website. This guy is extremely interesting. I’ll quote a bit from the Shermer review.

Bloom, it seems is big on metaphor.

Shermer’s review of Bloom’s; Global Brain
How can one capture the evolution of everything in the cosmos from the start to now in a single book? One way is through metaphor, and Bloom’s choice for his carrier is the computer—more specifically, the Internet and World Wide Web—that he hopes will transfer the idea of nerve cells communicating across a brain to individuals talking across a world.

Bloom correctly credits the metaphor to others (global brain metaphors have been common since the early 1980s), but he sees something deeper, in both time and space. “This planetary mind is neither uniquely human nor a product of technology.” ....

Bloom’s “new scientific theory,” as he calls it, explains “the inner workings of something to which conventional evolutionary thinkers have been blind: a planet pulsing with a more-than-massive data-sharing mind.” Why haven’t these scientists shared Bloom’s vision? The tyranny of individual selection has blinded them to the possibilities of group selection. This is a contentious issue tantamount to, if you will excuse my own metaphor making, Baptists and Anabaptists debating the merits of infant baptism, with emotions running just as high and factions fighting just as divisively. .....

Such mass metaphorical making, interdisciplinary thinking is at the heart of Bloom’s weakness as a thinker; it is also, and undeniably, his greatest strength. I am intrigued by the unique intellectual style of Bloom, a one-time music magazine publisher and rock promoter, who coupled his interest in social relations to his background in science to generate a number of interesting observations and deductions in Global Brain. Despite my reservations, this is a clever book, meticulously researched, beautifully written, and well worth reading, even if your neural nets do not hyperlink the data to other neural units in your broadbanded connections.

Here are a few of the logic gaps in this idea:
the integer elements of primes having the least “entropy” thing, whilst interesting, bears no real parallel with actual entropy and partition functions are just a simplification rather than the full story. Besides which, there is no reason in any real system to reduce the partition functions into integers rather than any other real numbers

quantum superposition only works if the quanta are in the same location, which not all of the quanta in the universe are, plainly.

it bears no real similarity to an eratosthenes sleeve, since the probability densities that you obtain for multiple experiments to determine a property of a single quantum give wavelike plots that are asymmetric.

maximum entropies existing as gradient and area calculations are no evidence whatever of interelation in the first place, and in the second, given that I can see points at which the differential is zero and others at which it is in between maximum and minimum in your diagram this is fallacious anyway. Thirdly, I can see regions where you can set limits that would give in-betweeny integrals. So even if a real quantum system could be represented by an eratosthenes sleeve (which it can’t), your idea of non-local and local inrelation is flawed.

if we expand the wave particle duality to a whole universe? how? and in what way? wave particle duality really only works on the basis of a quantum being standing wave like and therefore having a region over which it operates, but also particle like in that it has a defined focus of maximum amplitude at which a property can be said to be acting. this last point varies according to where the quantum itself phically moves to and how the energy is being transferred along it from one instant to the next. as you increase the number of quanta that make up an object, the less wavy that objects behaviour is and even at the size of an atom, this borders on negligible. so a whole universe is barely affected by the probabilistic wave-like properties of individual quanta.

the idea that an observer affects the outcome in quantum experiments is nothing more than a poetic notion that still persists in the minds of a very few serious scientists and the idea that ethics affect the outcome doesn’t actually follow from anything you have said and in purely spurious in its introduction here.

I don’t really see how your pretty little fractal models follow from any of the foregoing discussion. there is no link given and I strongly expect no link is ascertainable.

In short, you seem to take the erroneous premise that primes having the lowest entropy if you only use integers despite their being no reason why one should, then ignore it for the rest of the discussion whilst going to the erroneous premise that the eratosthenes sleeve resembles any probabilistic quantity associated with quanta, use this to make a massive leap of faith to say that locality and non-locality are interrelated despite the fact that it would a) depend on which values you chose for your y-axis and b) their clearly not and one of which is a divergent quantity anyway, and then you spuriously add that ethics effects quanta and go onto add a whole section in which pretty little fractals are generated to no purpose.